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Properties

Label 2-1344-21.11-c0-0-1
Degree $2$
Conductor $1344$
Sign $-0.0633 + 0.997i$
Analytic cond. $0.670743$
Root an. cond. $0.818989$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + 13-s + (−0.5 − 0.866i)19-s − 0.999·21-s + (−0.5 + 0.866i)25-s − 0.999·27-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (0.5 − 0.866i)39-s + 43-s + (−0.499 + 0.866i)49-s − 0.999·57-s + (1 + 1.73i)61-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + 13-s + (−0.5 − 0.866i)19-s − 0.999·21-s + (−0.5 + 0.866i)25-s − 0.999·27-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (0.5 − 0.866i)39-s + 43-s + (−0.499 + 0.866i)49-s − 0.999·57-s + (1 + 1.73i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.0633 + 0.997i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :0),\ -0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.165984704\)
\(L(\frac12)\) \(\approx\) \(1.165984704\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - 2T + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.383467158233869330309434519483, −8.808915666656241617420891102977, −7.86080495463675631444327463042, −7.19652795889189431779836154334, −6.46804651908342530034393502210, −5.68546210133623193923663978823, −4.20038556684155557208127900518, −3.45768450465414803479217889066, −2.32619378445995974936277107524, −0.971873684487282232698546411294, 2.00816874465665383006964055389, 3.11227141670397092014620725843, 3.88051996524180013920209145277, 4.91505695290780064439623235451, 5.86860027522183108490775221435, 6.53153640417336782669873061070, 7.960070583648503875698738693212, 8.506554621931060771096713992486, 9.170360850273264025598008120593, 9.997954344333303524717599043405

Graph of the $Z$-function along the critical line